Question

Find the value of the expression ${\sum }_{i=1}^n{\sum }_{j=1}^i{\sum }_{k=1}^{j}1$

Answer 1

${\sum }_{i=1}^n{\sum }_{j=1}^i{\sum }_{k=1}^{j}1={\sum }_{i=1}^n{\sum }_{j=1}^{i}j$
$-\sum _{i=1}^n\frac{i(i+1)}{2}=\frac{1}{2}[\sum _{i=1}^n{i}^2+\sum _{i=1}^{n}i]$
$=\frac{1}{2}[\frac{n(n+1)(2n+1)}{6}+\frac{n(n+1)}{2}]$
$=\frac{n(n+1)}{10}[2n+1+3]=\frac{n(n+1)(n+1)}{6}$